Methods, apparatus and systems for real time identification and control of modes of oscillation

ABSTRACT

A system for real time identification of modes of oscillation includes a sensor, an observer, a controller and an actuator. The sensor senses a controlled system such as a combustor, and generates a signal indicative of the modes of oscillation in the controlled system. For example, these modes of oscillation can be combustion instabilities. The observer receives the signal from the sensor, and uses the signal to determine modal functions and frequencies of the modes of interest with a pair of integrals with changing time limits. The controller receives the modal functions and frequency for each mode of interest from the observer, and effects a gain and phase shift for each mode. Based on the modal functions, the frequency, the gain and the phase shift, the controller generates and outputs a control signal, that is supplied to the actuator. The actuator controls the modes of oscillation of the controlled system, based on the control signal. The system of this invention can be used to damp or enhance oscillation modes of the controlled system, depending upon whether the oscillation modes are beneficial or detrimental to system performance.

STATEMENT OF RIGHTS TO INVENTION MADE UNDER FEDERALLY-SPONSORED RESEARCHAND DEVELOPMENT

This invention was developed pursuant to A.F.O.S.R. Grant No.F49620-93-1-0177. Accordingly, the U.S. Government has a paid-up licensein this invention.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates primarily to the field of real timecharacterization and control of quasi periodic or periodic signals.Subjects in this field that can benefit from the invention includesignal identification in a noisy background, coherent jammingidentification and general real time signal identification. Morespecifically, the invention has been developed in conjunction with thefield of vibration and oscillation control. This field covers subjectssuch as vibration damping in structures and machines, active noisereduction, control of compressor surge and stall, unsteady combustioncontrol and unstable combustor control. The invention is particularlysuitable for active control of combustion instabilities where it canrapidly identify and prevent the onset of detrimental combustioninstabilities before they damage the system and/or prevent the systemfrom attaining its objectives.

2. Description of the Related Art

Detrimental combustion instabilities, characterized by large amplitudeoscillations of one or more natural acoustic modes of the combustor,often occur in propulsion systems, such as rocket motors, ramjets, airbreathing jet engines, and gas turbines that are used for powergeneration and industrial combustion systems. A combustion instabilityoccurs when energy supplied by the combustion process excites naturalacoustic modes of the combustor via a feedback process between theacoustic and heat addition oscillations. The combustor oscillations thataccompany the onset of combustion instability must be prevented orsuppressed immediately after their appearance to avoid severely damagingor destroying the system, and destructive interference with the system'smission.

To date, passive approaches have been mainly used to prevent theoccurrence and/or reduce the amplitude of combustion instabilityoscillations. A passive approach may involve a combination of one ormore of the following: modification of the combustion process, changingthe combustor geometry, and addition of damping elements to the system.Modifications of the combustion process, such as changing thefuel/oxidizer ratio or reactants feed system design, aim to reduce theamount of energy supplied by the combustion process to the oscillations.Changes to the combustor geometry, for example, by welding baffles tothe injector face of a liquid propellant rocket motor, aim to limit thenatural acoustic modes that can be excited in the combustor, and thusprevent excitation of unstable modes. Damping elements, such asHelmholtz resonators, are added to the system to increase its acousticdamping and prevent the excitation of potentially unstable modes orreduce their amplitudes to low levels. Unfortunately, due to lack ofadequate understanding of the processes that control combustioninstabilities, the implementation of passive control approachesgenerally requires lengthy and costly development programs, which oftenfail to attain their objectives. Moreover, these solutions are notuniversal and a different cure must be found whenever a new instabilityarises. Consequently, new approaches for controlling detrimentalcombustion instabilities are needed.

The shortcomings of passive control can be overcome by an active controlapproach. An active control system generally consists of a sensor, acontroller and an actuator. A sensor such as a pressure transducer or aphoto-multiplier is used to measure the combustor pressure or radicalradiation, respectively. The sensor sends the measured signal to thecontroller where it is analyzed, modified and sent to the actuator whosetask is to modify the operating conditions within the combustor in amanner that prevents the onset or causes rapid attenuation of theinstability. The actuator may be a speaker that excites pressureoscillations within the combustor, an oscillating valve thatperiodically varies the air flow rate into the combustor or an injectorthat modulates the fuel and/or oxidizer flow rate into the combustor. Inorder to damp the instability, the actuator must introduce disturbanceswithin the combustor that tend to attenuate the instability.

Current active control methods are based upon time domain approaches.Such time domain approaches typically assume that the instabilityconsists of a single frequency oscillation. The measured signal istransmitted to a compensating network where it is provided with anappropriate amplification and phase shift. The modified signal is thensent to an actuator that excites a nearly pure harmonic oscillationwithin the combustor that is out of phase with the combustioninstability oscillations. This approach was demonstrated in simplelaboratory combustors, such as a Rijke burner, where single, lowfrequency longitudinal mode oscillations were excited. Langhorne, P. J.,Dowling, A. P. and Hooper, N., Journal of Propulsion and Power, 6(3),324, 1990. Such experiments represent a considerable simplification ofwhat happens in practical combustors where complex oscillationsconsisting of several modes, each characterized by a different frequencyand amplitude, are excited. Most importantly, the characteristics of theexcited oscillations are generally not known in advance. To overcomethis problem, Billoud, G., Galland, M. A., Huynh Huu, C. and Candel, S.,Combust. Sci. and Tech., 1992, Vol. 81, pp. 257-283, used an adaptivefilter whose function was to assure that the utilized compensationnetwork introduced proper amplification and phase shift as the frequencyof the instability changed. Unfortunately, the implementation of such anadaptive process is inherently very slow and, thus, cannot rapidlysuppress detrimental combustion instabilities.

Y. Fung and V. Yang, "Active Control of Nonlinear Pressure Oscillationsin Combustion Chambers,", 27th AIAA/ASME/SAE/ASEE Joint PropulsionConference, Sacremento, Calif., 1991, advocate the use of a more globaltime domain approach, based upon the state feedback method, in thecontrol of unstable combustors. This method describes the overall stateof a system by a minimal number of state variables that vary with time.The system is controlled by a feedback loop that depends upon the statevariables. In many cases, the state variables cannot be fully measuredand an observer is used to estimate the state of the controlled system.The state variables feedback approach is most suitable for systems thatcan be described by a finite number of ordinary differential equations.Such an approach is, however, not suitable for describing the state ofsystems, such as an unstable combustor, whose behavior is described bypartial differential equations. This problem can be handled in simplecases, such as axial instabilities, by approximating the partialdifferential equations that describe the behavior of the system by anensemble of an infinite number of ordinary differential equations usingorthogonal mode expansions, such as the Galerkin method. Zinn, B. T. andLores, M. E., Combustion Science and Technology, 4, 269, 1972, Culick,F. E. C., "Combustion Instabilities in Liquid Fueled PropulsionSystems-An Overview", California Institute of Technology Report, 1988.Since it is impossible to control a system that is described by aninfinite number of ordinary differential equations, it is furtherassumed in such cases that the behavior of the system, in the relevantfrequency range, can be described by a finite number of ordinarydifferential equations that is a subset of the original system ofinfinite differential equations. A difficulty is encountered in thisapproach because the determination of the state of the system requiresthe installation of many sensors along the combustor, which is obviouslyimpractical. Y. Fung and V. Yang "Active Control of Nonlinear PressureOscillations in Combustion Chambers," 27th AIAA/ASME/SAE/ASEE JointPropulsion Conference, Sacremento, Calif., 1991, suggested that thisproblem be handled in combustors by measuring the pressure at onelocation and using an observer to estimate the spatial dependence of thepressure with the aid of a model of the system. This approach requires,however, full knowledge of the complex fluid mechanical and combustionprocesses within the combustor, which is beyond current state of the arteven for the simple case of longitudinal instabilities, let alone themore complex radial, transverse and three-dimensional instabilities,which often occur in actual combustors. The difficulties encountered byproposed active control systems for unstable combustors clearly indicatea need for a new active control approach.

In contrast to the detrimental effects produced by the excitation ofnatural acoustic mode oscillations in propulsion and power generatingsystems, the excitation of finite amplitude acoustic mode oscillationsin practical combustors, such as incinerators, and energy intensiveprocesses, such as dryers and boilers, produces such benefits as fuelsavings, increased productivity, reduced emissions and improved productquality. In U.S. Pat. No. 4,699,588, Zinn, et al. have shown thatbeneficial resonant acoustic oscillations can be excited in industrialprocesses by use of tunable pulse combustors. Yet, alternate apparatusesand methods for exciting such acoustic oscillations, that require lessspace and capital investment and are easier to operate, are needed. Theinvention described herein can be used for these purposes.

SUMMARY OF THE INVENTION

The present invention provides a novel approach for active control ofdetrimental instabilities in practical combustors or the excitation ofbeneficial acoustic oscillations in combustion and energy intensiveprocesses, for example. The present invention overcomes theabove-described and other problems in prior active control systems. Theinvented system comprises a sensor, a processing unit including anobserver and a controller, and an actuator. The observer analyzes thesignal supplied by the sensor and determines, virtually in real time,the modal functions (and hence the amplitudes and phases) andfrequencies of the excited combustor modes. Based on the modal functionsand frequencies, the controller adaptively determines a gain and phaseshift for each mode or uses a predetermined gain and phase shift, andcombines the modes to generate a time-varying control signal that is fedto the actuator. Based on the control signal, the actuator produces asecondary system of oscillations within the combustor that tends to dampthe instability. Alternatively, the controller can assign a phase and again to the actuator signal in a manner that excites oscillations in theprocess for applications where such oscillations produce beneficialeffects.

The observer of this invention uses a frequency domain analysis thatavoids the time consuming process of frequency scanning commonly used infast Fourier transform (FFT) and Wavelets transformers. Instead, theunknown frequencies and amplitudes of the modes of oscillation aredetermined by the processor using a pair of integrals that are modifiedversions of Wavelet Transforms, whose integration limits are changedcontinuously. The frequency of the largest amplitude, dominant mode isdetermined first in a rapidly converging solution approach. Once thedominant mode has been determined, it is effectively subtracted from thesensor signal as are the other modes as presently known by the observer,and the mode characterization procedure is repeated to determine thecharacteristics of the mode with the next largest amplitude. Modecharacterization as described above is repeated until all modes orinterest have been characterized by determining respective modalfunctions and frequencies. The amount of processor calculations requiredfor each mode identification is very small and comparable to that neededto perform the simplest first order filtering in time domain. The smallcomputational load required by the invention permits the observer to beimplemented with a standard micro-processor or micro-controller toanalyze, virtually in real time, the characteristics of oscillationswith relatively high frequencies up to 5,000 Hz, for example.

An actuator in accordance with this invention is implemented as aninjector system for controlling a fluid (e.g., oxidizer fuel) flow intoa combustor, engine or the like. The injector system uses amagnetostrictive actuator capable of responding to a control signal withrelatively high frequency components. The injector system modulates thefuel flow rate into the combustor to provide highly effective excitationof oscillations with desired characteristics within the system. Theinjector system thus can be used to attenuate or excite oscillationswithin the controlled system. Also, in accordance with this invention,drift effects of the injector system caused by hysteresis or heating,are compensated using a separate controller.

These together with other objects and advantages, which will becomesubsequently apparent, reside in the details of construction andoperation as more fully hereinafter described and claimed, referencebeing made to the accompanying drawings, forming a part hereof, whereinlike numerals refer to like parts throughout.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention can be better understood with reference to thefollowing drawings. The drawings are not necessarily to scale, emphasisinstead being placed upon clearly illustrating the principles of theinvention.

FIG. 1 is a general block diagram of a system for active control ofoscillations, in accordance with the present invention;

FIG. 2 is a block diagram of a processing unit that can be used torealize the observer and/or controller of this invention;

FIGS. 3A and 3B are views of respective wavelets;

FIG. 4 is a flow chart of processing used to identify the modalfunctions and frequency of modes of oscillation, in accordance with thepresent invention;

FIGS. 5A, 5B and 5C are flow charts of processing performed by theobserver to identify modal functions and frequencies of modes ofoscillation in a sensor signal, in accordance with the presentinvention;

FIGS. 6A and 6B are flow charts of processing performed by thecontroller to determine the optimal phase and minimum gain required forsuppressing undesirable oscillations, in accordance with the presentinvention;

FIG. 7 is a cross-sectional view of a fast response actuator and a driftactuator in accordance with this invention; and

FIG. 8 is a block diagram of a controller for the drift actuator inaccordance with this invention.

DESCRIPTION OF TEE PREFERRED EMBODIMENT

FIG. 1 is a block diagram of a system 1 of this invention applied to asystem to be controlled, specifically, a combustor 2. As previouslynoted, the system 1 of this invention can be used in many applications,but for ease in understanding the principles of this invention, thefollowing description will be given for a specific application, i.e., acombustion system. However, extensions of this invention to otherapplications will be readily understood from the following description,and these extensions are to be considered as included in this invention.

In FIG. 1, the combustor 2 receives and burns fuel, which can cause acombustion instability. As previously explained, a combustioninstability exists when energy supplied by the combustion processexcites natural acoustic modes of the combustor 2 through a positivefeedback process between the combustion heat addition and acousticoscillations. Depending upon the purpose to which the system 1 isapplied, it can suppress detrimental combustion instabilities or excitebeneficial combustor oscillations. For example, in applications such asdryers or boilers, it may be desired to promote combustion processoscillations that produce fuel savings, increased productivity, reducedemissions and enhanced product quality. On the other hand, inapplications such as rocket motors, ramjets, afterburners and gasturbines, these combustion instabilities must be eliminated quickly toavoid damaging or destruction of such combustors.

Whether combustor oscillations are to be promoted or attenuated for aparticular application, the combustor oscillations must be sensed andcharacterized. To sense a combustor oscillation, a sensor 3 is arrangedin contact with and/or in proximity to the combustor 2. For example, thesensor 3 can be a pressure transducer, a transducer sensing vibrations,or a photomultiplier arranged near a window in the combustor 2, thatsenses the total amount of radiation or the radiation at a specificwavelength from the flame in the combustor 2. Of course, the sensor 3must have a frequency response sufficient to permit detection of thehighest oscillation mode frequency of interest in the combustor.

The sensor 3 is coupled to a processing unit 4. The processing unit 4includes an observer 5 and a controller 6. The observer 5 receives asignal generated by the sensor 3 and characterizes the combustoroscillations indicated by the received signal. More specifically, theobserver 5 determines, virtually in real time, the modal functions (andhence the amplitude and phase) and frequency of each significant mode ofoscillation included in the received signal. The observer 5 supplies thecontroller 6 with the modal functions and frequency of each mode ofoscillation. The controller 6 uses the modal functions and the frequencyof each mode of oscillation to synthesize a time-dependent controlsignal with an appropriate gain and phase shift for each identifiedmode. The gain and phase shift for each mode of oscillation can bepredetermined for the system, or if needed, can be determined by aprocess in which the controller 6 continually adjusts the gain and phaseshift of each mode of oscillation as necessary to optimize the dampingor enhancement of modes of oscillation, depending upon the objective ofthe system.

The controller 6 is coupled to output its time-dependent control signalto an actuator 7. The actuator 7 can be realized in several ways anddepends upon the system to be controlled. For example, the actuator 7used to control the combustor 2, can be a speaker that excites pressureoscillations within the combustor 2, based on the control signal. Also,the actuator 7 can be an oscillating valve that periodically varies theair flow rate into the combustor 2, based on the control signal.Further, the actuator 7 can be an injector that modulates the fueland/or oxidizer flow rate into the combustor 2, based on the controlsignal. A specific embodiment of the actuator 7 of the injector-type inaccordance with this invention, will be described later in this documentwith respect to FIGS. 6 and 7. In any case, based on the control signal,the actuator 7 acts to dampen or promote oscillations in the combustor2, depending on the objective of the system.

FIG. 2 is a block diagram of one possible embodiment of the processingunit 4. In FIG. 2, the processing unit 4 includes a processor 8 and amemory 9 coupled to a common bus 10. The processing unit 4 also includesan input/output (I/O) unit 11 coupled to the bus 10. The I/O unit 11 isalso coupled to receive the signal from the sensor 3, and coupled tooutput the control signal generated by the processor 8 and the memory 9,to the actuator 7. In FIG. 2, the processor 8, the memory 9, the bus 10,and the I/O unit 11 perform the functions of both the observer 5 and thecontroller 6 in FIG. 1. However, if desired, the observer 5 and thecontroller 6 can be realized as separate micro-controller orprocessor-based units, for example.

Before proceeding to the specific processing performed by the observer 5and the controller 6, an explanation of the theory from which theprocessing of the observer 5 has been derived will be helpful tounderstand the invention.

Consider the two modal functions a(t,ω) and b(t,ω) obtained from asignal f(t) by the following transformation ##EQU1## in which Ψ_(a) andΨ_(b) establish an orthogonal pair of wavelet functions localized aroundthe frequency ω and a time window between t-T and t, where T=2π/ω (forbasic information concerning Wavelet transformations, see "Ten Lectureson Wavelets," Ingrid Daubechies, Society for Industrial and AppliedMathematics, Philadelphia, Pa., 1992). While in general, various waveletfunctions may be used in conjunction with the present invention, thepair selected for the preferred embodiment is shown in FIGS. 3A and 3B.Specifically, the wavelet functions are sinωt and cosωt that are notidentically zero only during the time interval from t-T to t. With thispair of wavelet functions, the integrals in Eqs. (1) and (2) assume thesimple form in which the time localization appears explicitly in theintegral limits. ##EQU2##

Consider now a reverse transformation that reconstructs a signal y(t)from the modal functions a(t,ω) and b(t,ω) determined by formulas (3)and (4) as follows:

    y(t,ω)=a(t,ω)sinωt+b(t,ω)cosωt (5)

Equations (3), (4) and (5) define a transformation f(t)→y(t,ω) from thetime domain into time and frequency domain.

The transformation defined by Eqs. (3), (4) and (5) possesses thefollowing qualities:

The transformation f(t)→y(t) is linear; that is, if f₁ (t)→y₁ (t) and f₂(t)→t₂ (t) then Uf₁ (t)+Vf₂ (t)→Uy₁ (t)+Vy₂ (t) where U and V areconstants.

II. If the input f(t) oscillates with a frequency Ω, the output signaly(t) also oscillates with the same frequency Ω even if the frequency ω,which is used in y(t), differs from the actual frequency Ω; that is,ω‡Ω.

III. The frequency of the input signal can be determined analyticallyfrom the above expressions for the coefficients a(t) and b(t) using theformula ##EQU3## where a and b are the derivatives of a(t) and b(t),respectively, with respect to time.

Note that the reconstructed signal is a function of the frequency ω eventhough it oscillates with the same frequency Ω as the original signalf(t). This is so because the reconstructed signal possesses a phase andattenuation with respect to the signal f(t) that depends upon thedifference between the actual frequency Ω and the estimated frequency ω.Thus, for full recovery of the signal f(t), the actual frequency Ω mustbe determined. Wavelet or Windowed Fourier transforms require scanningof many frequencies to capture the oscillation modes in a signal. Forthis reason Wavelet and Fourier transformers are not adequate for signalprocessing in real time control applications. In the observer of thisinvention, however, Equation (6) is used to update the lower limit ofthe integral and thus in fact to update the wavelet functions Ψ_(a) andΨ_(b). The adaptation of the limits of the integral leads to aconverging process in which the estimated frequency ω approaches theactual frequency Ω. For this reason, the dependence of the modalfunctions a(t) and b(t) upon a particular frequency ω becomes negligibleand, thus, may be considered as functions of time alone. The convergingtransformation possesses the following important quality.

IV. When the input signal f(t) contains several modes of oscillationthat oscillate with different frequencies, the observer inherentlyidentifies the frequency of the largest amplitude mode of oscillationfirst.

To determine the modal content of the input signal, the observer modecharacterization processing is repeated for each mode until thecharacteristics of all modes of interest have been determined.Hereinafter, the completion of the characterization of all modes ofinterest will be referred to as an `iteration` of the processingperformed by the observer 5.

A simplified example of the successive characterization of modes ofoscillation is given below. In the first iteration of this process, themode with the largest amplitude (that is, the largest magnitude of modalfunctions) is characterized using a pair of integrals with variablelimits, and output as the observer output signal y(t). As will beexplained in more detail with respect to FIGS. 4, 5A, 5B and 5C, thedominant mode is then effectively subtracted from the sensor signalalong with predetermined initial estimations of the other modes ofinterest, to generate a new modal input signal f(t) that is provided tothe observer. This modal input signal is then analyzed by the observerusing equations derived from modified versions of integrals withtime-varying limits, to generate another observer output signal. The twocharacterized modes are then effectively subtracted from the signalreceived from the sensor 3, along with the initialized modes as yet tobe determined by the observer, to generate another modal input signalfor the observer. The processing performed by the observer 5 continuesin the above-described manner until all modes of oscillation have beencharacterized in one iteration. In the next iteration, the time isstepped (i.e., t=t+dt) and the above-described procedure is repeated.Starting with initial values, the modal output signals are improved witheach iteration, until they converge to the actual modes of oscillationin the controlled system.

Because of the large numerical effort, determination of the integralsdescribed in Eqs. (3) and (4) is not practical for real time processing.However, this invention uses recursive formulae that substantiallyreduce the numerical efforts required for the evaluation of theintegrals in Eqs. 2 and 3. These recursive formulae are derived fromEqs. 2 and 3 as follows: ##EQU4## Equations (7) and (8) are solved bythe observer 5 using the following procedure. First, a data array ofshort time intervals or increments Δt whose sum equals one estimatedperiod of the mode of oscillation T is established. The number ofelements in this array equals T/Δt, where Δt is the time step for eachiteration. Each time step in the cycle corresponds to a differentlocation in the array. Thus, a cycle or period starts at the firstelement and ends at the last element in the array. According to Eqs. (7)and (8), the modal functions a(t+dt) and b(t+dt) at time t+dt aredetermined by multiplying the difference between the current value ofthe modal input signal f(t+dt) supplied to the observer, and the modalinput signal f(t-T+dt) supplied to the observer, obtained a period ofthe mode of oscillation earlier, by (2/T)sin(ωt) or (2/T)cos(ωt) andadding the result to the modal functions a(t) or b(t) at time t,respectively. This procedure substantially reduces the numerical effortrequired for the determination of the modal functions a(t) and b(t) (seeEqs. (1) and (2)) of the observed mode within the modal input signal.

The period of the observed mode generally varies during the calculationdue to actual changes in the process and/or an initial error in theassumed frequency. Consequently, the length of the data arraycorresponding to a period of the assumed frequency, also changes. Unlesscorrective steps are taken, this frequency variation may introduce anerror into the computed modal functions a(t) and b(t). This error is forpractical purposes eliminated in the processing used by the observer 5by the following procedure. In addition to the continuous computationsof the modal functions a(t) and b(t), using Eqs. (7) and (8), the valuesof A(T) and B(T) at the end of each period T are determined by use ofthe following equations: ##EQU5## where t_(o) denotes the beginning of aperiod. The values of A(T) and B(T), calculated using Eqs. (9) and (10),are used as the initial values for the calculation of the modalfunctions a(t) and b(t) using Eqs. (7) and (8), at the beginning of thefollowing cycle. Using this approach, the instantaneous variations ofthe modal functions a(t) and b(t) are determined by continuouslycalculating their values by use of Eqs. (7) and (8) and long term erroraccumulation is eliminated by replacing the instantaneous values of themodal functions a(t) and b(t) with the average values A(T) and B(T)calculated using Eqs. (9) and (10).

FIG. 4 is a flow chart of the processing performed by the observer 5 todetermine the modes of oscillation present in the sensor signal. Theflow chart of FIG. 4 is a generalization of the processing describedpreviously. The observer determines a modal input signal f₁ by weightedsummation of the sensor signal and modal output signals y₁, y₂, . . . ,y_(L) using initial values for the modal functions and the frequencyaccording to the equation y_(j) =a_(j) sin(ω_(j) t)+b_(j) cos(ω_(j) t)where j is the mode index assuming values from 1 to L. L is the totalnumber of modes of oscillation to be analyzed by the observer 5, and canbe predetermined and set by the user before beginning the processing ofFIG. 4. In step S1 of FIG. 4, in the first iteration, the observer 5generates a modal input signal f₁ by effectively subtracting from thesignal received from the sensor (in the general case, the receivedsignal can be weighted by a predetermined constant c₁) a weighted sum ofall L modes of interest c₁,1 y₁ to c₁,L y_(L) using initial values ofthe modal functions and frequencies for the L modes and predeterminedvalues for the constants c₁,1 to c₁,L. In step S2, in the firstiteration, the observer 5 characterizes (i.e., determines the modalfunctions and the frequency) the most dominant mode of oscillation ofthe received signal. The observer generates a modal output signal y₁based on this characterization that is stored in the memory 9 for use inthe next iteration. In step S3, the initialized modal output signals y₁,y₂, . . . , y_(L), weighted by multiplication with the coefficientsc₂,1, c₂,0,, . . . , c₂,L, are subtracted from the signal received fromthe sensor multiplied by a gain factor c₂, to generate a modal inputsignal f₂. In step S4, the observer 5 uses the modal input signal f₂ tocharacterize the second mode of oscillation. The processing performed bythe observer 5 continues in this manner until modal functions andfrequencies for all L modes of interest have been determined in thefirst iteration. In the next iteration, the modal functions andfrequencies of the modes determined in the previous iteration are usedto determine modal input signals for each mode. With each iteration ofthe processing of FIG. 4, the observer 5 generates modal output signalsy₁, y₂, . . . , y_(L) that are closer to the actual modes of oscillationof the sensor signal.

FIGS. 5A, 5B and 5C are flow charts of processing performed by theobserver 5 to characterize oscillation modes included in the signalreceived from the sensor 3. This processing can be performed, forexample, by the processor 8 under control of a control program stored inthe memory 9 in FIG. 2, using sample data obtained from the signalreceived from the sensor 3 by the I/O unit 11.

Before processing starts in FIG. 5A, the number of oscillation modes Lto be characterized and the maximum permitted length of a data arrayN_(MAX) are predetermined and programmed into the memory 9 (FIG. 2) sothat the processor 8 has access to these values. In FIG. 5A, processingstarts in step S11. In step S12, the processor 8 sets the mode indexj=1. The mode index j is used by the processor 8 to identify the mode ofoscillation to which various parameters pertain. The mode index jassumes values ranging from 1 to L, depending upon the mode ofoscillation under analysis. In step S13, initialization of variousparameters is performed. Specifically, the average values of modalfunctions over a period of the jth mode of oscillation A_(j), B_(j) andthe instantaneous values of the modal functions a_(j), b_(j) are set tozero. Recall that the values A_(j), B_(j) are used to replace theinstantaneous values of a_(j), b_(j) at the end of each period of themode of oscillation j. Also, in step S13, the output of mode j, y_(j),is set to zero. Further, the observed frequency of the jth mode, ω_(j),is set to a predetermined value. This predetermined value is merely aninitial estimate of the frequency of the jth mode of oscillation. Theinitial estimate ω_(j) can be predetermined and programmed by anoperator or an external device. In addition, the length of the array ofthe jth mode, N_(j), is set to 2π/(ω_(j) Δt). Finally, with respect tostep S13, the location index in the jth mode data array, k_(j), is setto zero.

In step S14, an index i used for initialization, is set to 1. In stepS15, the value at the location i of the data array, f^(old) _(i), is setto zero. In step S16, the processor 8 determines whether the index iequals the maximum size of a data array N_(MAX). If not, the index i isincremented by 1 in step S17 and processing returns to step S15. If theindex i is equal to the maximum size of a data array, N_(MAX), allvalues in the data array f^(old) _(i) of mode j have been initialized tozero and processing proceeds to step S18. In step S18, a determinationis made to establish whether j=L. If not, j is incremented by 1 in stepS19 and processing proceeds to step S13 to continue initialization ofthe parameters and data array f^(old) _(i) for the jth mode. On theother hand, if j equals the number of modes of oscillation to beobserved L, initialization of the parameters and data array has beenperformed for all L modes so that the processor 8 in conjunction withthe memory 9 and the I/O unit 11, are now ready to determine a modalinput signal and characterize a mode of oscillation. Accordingly,processing proceeds to step S20.

In step S20 of FIG. 5B, the processor 8 causes the I/O unit 11 to samplethe signal received from the sensor 3. The resulting data sample isstored in the memory 9. In step S21, the processor 8 sets the mode indexj equal to 1. This step is necessary if more than one mode ofoscillation is to be characterized because step S19 of theinitialization processing of FIG. 5A will cause j to assume a valuegreater than 1. In step S22, the processor 8 in conjunction with thememory 9 calculates the following parameters: ##EQU6##

    t=k.sub.j Δt

    y.sub.j =a.sub.j sin(ω.sub.j t)+b.sub.j cos(ω.sub.j t)

    ΔA=f.sub.j sin(ω.sub.j t)Δt

    ΔB=f.sub.j cos(ω.sub.j t)Δt

    Δa=2/T (f.sub.j -f.sup.old.sub.kj)sin(ω.sub.j t)!Δt

    Δb=2/T (f.sub.j -f.sup.old.sub.kj)cos(ω.sub.j t)!Δt

    a=Δa/Δt

    b=Δb/Δt.

The coefficients c_(j) and c_(i),j are predetermined values stored inthe memory 9 before starting the processing in step S10 of FIG. 5A.Also, the time interval Δt is predetermined and stored in the memory 9.The time interval Δt is predetermined in such a way that the timeinterval Δt is sufficiently long so that the processor 8 can perform aniteration of the processing of FIGS. 5B and 5C within the time intervalΔt, but is also predetermined to be short enough that as many samples aspossible of the signal received from the sensor, can be obtained over aperiod of the mode of oscillation with the highest frequency ofinterest. In step S23, a determination is made to establish whethera_(j) cosω_(j) t-b_(j) sinω_(j) t is greater than a predeterminedthreshold value. The predetermined threshold value is stored in thememory 9 before starting the processing of FIGS. 5A, 5B, 5C and is usedto prevent division by unreasonably small values in step S24 (note thatif the term a_(j) cosω_(j) t-b_(j) sinω_(j) t is small in thedenominator of the equation for Ω_(j) in step S24, the frequencyestimate ω_(j) may assume large, inaccurate values). If thedetermination in step S23 is affirmative, processing proceeds to stepS24. On the other hand, if the determination in step S23 is negative,step S24 is bypassed and processing proceeds directly to step S25.

In step S24, the frequency Ω_(j) is determined based on the followingequation: ##EQU7## Thus, the processor 8 uses the values ω_(j), a, b, t,a, b stored in the memory 9 to determine the frequency Ω_(j) whichdefines the frequency of the mode of oscillation with mode index j.

In step S25 of FIG. 5C, the processor 8 in conjunction with the memory 9calculates the following parameters: ##EQU8## Thus, the values A_(j),B_(j), a_(j) and b_(j) are updated by adding the changes in the valuesΔA, ΔB, Δa, Δb, respectively, over the time interval Δt (i.e., for oneiteration), to the former values of these parameters as initialized ordetermined in the previous iteration. The estimated frequency ω_(j) isalso updated. However, rather than setting the value ω_(j) =Ω_(j), ω_(j)is set equal to ω_(j) +(Ω_(j) -ω_(j))N_(j) so that the frequency ω_(j)is not changed radically to ensure smooth convergence to the frequencyΩ_(j) of the mode of oscillation under analysis. This updated frequencyω_(j) and the modal functions a_(j), b_(j) are output to the controller6 for determination of the control signal.

Because ω_(j) generally changes with each iteration, the number ofsamples N_(j) also may change. Thus, if ω_(j) is smaller than determinedin the previous iteration, more samples are needed. To provide thesesamples, not only is f^(old) _(kj) set equal to f_(j), but f^(old)_(Nj+kj) is also set equal to f_(j). Thus, in the event that ω_(j) has asmaller value than determined by the previous iteration so that the dataarray has more elements (i.e., values) than the previous iteration,there will be a relatively good estimate for data stored in elementsk_(j) at time intervals beyond the range of the N_(j) used in theprevious iteration. In step S26, a determination is made to establishwhether k_(j) equals N_(j). If kj equals N_(j), the processing performedby the processor 8 in conjunction with memory 9 proceeds to step S27 inwhich case k_(j) is set to zero, a_(j) is set to A_(j), b_(j) is set toB_(j), and thereafter both A_(j) and B_(j) are set to zero. Aspreviously explained, this operation provides a_(j) and b_(j) with goodestimates for the next iteration by setting these modal functions toA_(j) and B_(j), respectively. Because A_(j) and B_(j) are obtained byaveraging over the previous period of the frequency ω_(j) of the jthmode of oscillation, they are not subjected to long term erroraccumulation. In preparation for the next iteration, the values of A_(j)and B_(j) are set to zero. From step S27, processing proceeds to stepS28. If the determination in step S26 is negative, k_(j) is incrementedby one in step S29 and processing proceeds to step S28.

In step S28, the processor 8 in conjunction with the memory 9 determineswhether j equals L. If not, the mode index j is incremented in step S30and processing proceeds to step S22. On the other hand, if j equals L instep S28, processing proceeds to step S20 to obtain the next value ofthe sensor signal for the next time step Δt.

As previously noted, in step S24 of FIG. 5B, the observer 5 outputs thefrequency ω_(j) and modal functions a_(j), b_(j) to the controller 6. Ifthe observer 5 and the controller 6 are realized together as theprocessing unit 4 of FIG. 2, this output is unnecessary as the processor8 will have access to these values stored in the memory 9. Thus, thecontroller 6 needs only to use an appropriate gain factor G_(j) and thephase shift φ_(j) to be used for each mode in the synthesized controlsignal. The gain factor G_(j) and the phase shift φ_(j) for each modecan be stored in the memory 9 as predetermined values. In an applicationin which the gain factor G_(j) and the phase shift φ_(j) cannot bedetermined a priori, the modal output signals can be used to adaptivelyadjust the gain factor G_(j) and the phase shift φ_(j) in an iterativeprocess. An example of such an iterative process used to damposcillation modes is described below with respect to FIGS. 6A and 6B.

In FIG. 6A, processing performed by the controller 6 (which in this caseis the processor 8 in conjunction with the memory 9 and the I/O unit 11(see FIG. 2)) begins in step S101. In step S102, the mode index j is setto 1. This mode index j is not the same mode index j used in FIGS. 5A,5B and 5C, but serves a similar purpose, that is, to distinguish themodes to which various parameters pertain. Also, in step S102, the indexi is set to 1. The index i is an index corresponding to time. Thus, thisindex is different from the index i used in initialization in theprocessing of FIGS. 5A through 5C. In step S103, the gain G_(j) is setto zero, as are the phase values φ_(j), φ_(oj), and φ_(optj) for allvalues of the mode index j. The phase φ_(j) is the current value of thephase used to generate the control signal, φ_(oj) is the initial valueof the phase used to generate the control signal with the current gainG_(j), and φ_(optj) is the estimated optimal phase for the controlsignal.

In step S104, the controller 6 receives the coefficients a_(i),j andb_(i),j of mode j from the observer 5. If the observer 5 and thecontroller 6 are realized together as the single processing unit 4 ofFIG. 2, the observer 5 need not output the coefficients a_(i),j andb_(i),j of the mode j because the memory 9 storing these coefficients,is common to both the observer 5 and the controller 6. However, if theobserver 5 and the controller 6 are implemented as separatemicrocontroller- or microprocessor-based units, step S104 is performedto provide the controller 6 with the coefficients a_(i),j and b_(i),j ofthe mode j from the observer 5.

In step S105 of FIG. 6A, a determination is made to establish whetherthe mode index i is equal to 1. If so, processing proceeds to step S113.On the other hand, if the mode index i is not equal to 1, processingproceeds to step S6 of FIG. 6A.

In step S106, a determination is made to establish whether the modeamplitude or magnitude, (a_(i),j)² +(b_(i),j)² !^(1/2), is greater thana predetermined threshold value. This predetermined threshold value isstored in the memory 9 in advance of starting the processing of FIGS. 6Aand 6B. The determination in step S106 establishes whether the amplitudeof the jth mode of oscillation is sufficiently large that it should bedamped. If the determination in step S106 is negative, j is incrementedby 1 in step S107 and processing returns to step S103 in FIG. 6A. On theother hand, if the determination in step S106 is affirmative, processingproceeds to step S108 of FIG. 6B.

In step S108, the processor 8 in conjunction with the memory 9determines whether (a_(i),j)² +(b_(i),j)² !^(1/2) is greater than(a_(i),j)² +(b_(i-1),j)² !^(1/2). The determination in step S108therefore establishes whether the amplitude of the jth mode ofoscillation is increasing over a time step defined by the index i. Ifthe determination of step S108 is affirmative, processing proceeds tostep S109. On the other hand, if the determination in step S108 isnegative, processing proceeds to step S110. In step S109, adetermination is made to establish whether the gain G_(j) is equal tozero. If the gain G_(j) is zero, processing proceeds to step S11 inwhich the gain G_(j) is increased, for example, by a predeterminedincrement stored in the memory 9 in advance of performing the processingof FIGS. 6A and 6B. From step S111, processing proceeds to step S112where φ_(oj) and φ_(optj) are set equal to φ_(j), and control proceedsto step S113. On the other hand, if the determination of step S109establishes that the gain G_(j) does not equal zero, processing proceedsto step S114 in which φ_(optj) is set to φ_(j) +90°. Step S114 isperformed because, if the gain G_(j) is not equal to zero in step S109and the current magnitude of the present modal functions a_(i),j,b_(i),j are greater than the magnitude of the previous modal functionsa_(i-1), b_(i-1),j as determined in step S108, the phase φ_(j) used togenerate the control signal in step S116, must be incorrect by at least90°. Thus, step S14 is used to adjust the phase φ_(optj) for use in stepS110 where the phase φ_(j) is determined. In step S115, a determinationis made to establish whether φ_(j) -φ_(oj) is greater than 360°. If so,further adjustments of the phase φ_(j) to improve damping of the jthmode of oscillation will be fruitless because the gain G_(j) isinsufficient to damp the jth mode of oscillation. Thus, if thedetermination in step S115 is affirmative, processing proceeds to stepS111 to increase the gain G_(j) by a predetermined increment so that thejth mode of oscillation can be damped. On the other hand, if thedetermination in step S115 is negative, processing proceeds to stepS110. In step S110, φ_(j) is set equal to φ_(j) +Δt(φ_(optj) -φ_(j))/τ.The newly determined phase φ_(j) represents the new controller phase forthe jth mode of oscillation. Rather than radically changing the previousphase value φ_(j), the factor Δt/τ where τ is greater than Δt, is usedto gradually adjust the current phase φ_(j) to the estimated optimalphase φ_(optj). The time constant τ is predetermined and stored in thememory 9 before the start of the processing in FIGS. 6A and 6B as is thetime interval Δt (this Δt is not necessarily related to the timeinterval Δt of FIGS. 5A through 5C).

After step S110 or step S112 are performed, processing proceeds to stepS113 to determine if j=L (this L is the same as that L used in theprocessing of FIGS. 5A, 5B and 5C). If so, one iteration of phaseadjustments for all of the L modes of interest has been completed.Accordingly, in step S116, the controller 6 (i.e., the processor 8 inconjunction with the memory 9 and the I/O unit 11), generates andoutputs the control signal: ##EQU9## This control signal is output fromthe I/O unit 11 to the actuator 7.

After outputting the control signal in step S116, processing proceeds tostep S117 to step the time by incrementing the index i by 1. Afterperforming step S117, processing proceeds to step S118 to set the modeindex j equal to 1 preparatory to performing the next interaction of theprocessing of FIGS. 6A and 6B. After step S118 in FIG. 6B, processingproceeds to step S104 of FIG. 6A. Returning to step S113 of FIG. 6B, ifj is not equal to L in step S113, the phase and/or gain has yet to bedetermined for all L modes of oscillation. Accordingly, in step S119 ofFIG. 6B, the mode index j is incremented by 1 and processing proceeds tostep S104 of FIG. 6A.

The processing performed by the processing unit 4 in FIGS. 5A, 5B, 5Cand FIGS. 6A and 6B, provides damping of combustion instabilitiesvirtually in real time. With this invention, performance of systems suchas the combustor 2 can be enhanced well beyond the capabilities ofsystems developed before this invention. To more fully benefit from therapid characterization of oscillation modes and generation of anappropriate control signal made possible by the processing of FIGS. 5A,5B, 5C, and FIGS. 6A and 6B, a fast response actuator 7 is needed. Anembodiment of such an actuator is explained below with reference to FIG.7.

FIG. 7 is an embodiment of the actuator 7 realized as a fuel injector.Fuel is supplied to the injector through a supply port 12 defined by ahousing 13. From the supply port 12, the fuel moves through an orifice14 defined by the housing 13, and into an orifice 15 created between theinner wall of a tube 16 and outer wall of a tapered needle 17. Amagnetostrictive actuator 18 is coupled to reciprocate the taperedneedle 17 in an up-and-down motion in FIG. 7 in response to changes inan electric field input based on the control signal from the controller6. Preferably, the control signal is amplified by a relativelyhigh-frequency power amplifier (not shown) coupled between thecontroller and the magnetostrictive actuator 18 so that themagnetostrictive actuator receives an amplified version of the controlsignal. The movement of the tapered needle 17 causes changes in theeffective area of the orifice 15. Since the flow rate of the fuel isdirectly proportional to the magnitude of the orifice area 15, the timedependence of the fuel flow rate through the orifice 15 can be modulatedby periodic or quasi-periodic variation of the orifice area 15 based onthe control signal from the controller 6. This is accomplished bycontrolling the reciprocating motion of the needle 17 by appropriatelyvarying the electric input into the magnetostrictive actuator 18 basedon the control signal from the controller 6. The variation of the fuelflow rate through the orifice 15 results in a variable fuel injectionrate into the combustor 2 through the injection tube 19. A motion sensor20 is coupled to the needle 17 by a member 21. The motion sensor 20generates an output signal for monitoring the movement of the needle 17,and, if necessary for the particular application in which this inventionis used, for controlling its motion in a feedback manner with anappropriate controller (not shown) in a closed loop. A knob 22 is usedto set the mean position of the needle 17 relative to the opening at theend of the tube 16, to control the mean area of the orifice 15 and thusthe mean fuel flow rate through the injector. The flow rate through theinjector is determined by measuring the pressure drop across orifice 14using a differential pressure transducer 23. The differential pressuretransducer 23 generates an output signal indicative of the differentialpressure across the orifice 14. The output signal from the differentialpressure transducer 23 is used to control the mean area of the orifice15 with a drift actuator 24, as will be described below.

The magnetostrictive actuator 18 provides the injector with a fastresponse time, a large actuation force and appreciable displacementamplitude. The use of a magnetostrictive actuator offers designsimplicity, rapid response time and actuation force that are superior tothose offered by most solenoid motors and actuators. However, acontroller is needed to compensate for the large hysteresis of theTerfenol-D™ material that comprises the core of the magnetostrictiveactuator 18 (for example, such as the magnetostrictive actuatormanufactured by Etrema Products™, Inc. of Ames, Iowa), its significantthermal expansion and variation of operational characteristics withtime. In accordance with this invention, a controller 25 for themagnetostrictive actuator 18 is described below with reference to FIG.8.

In FIG. 8, the drift of the magnetostrictive actuator 18 is controlledby the controller 25. The controller 25 includes an adder 26 with anoutput coupled to a low pass filter 27. As inputs, the adder 26 receivesthe output signal indicative of the flow rate from the differentialpressure transducer 23 that measures the pressure drop across theorifice 14. The adder 26 subtracts the output signal of the differentialpressure transducer 23 from a predetermined set point of the mean flowrate, which is set externally to the system 1 of this invention. Theadder 26 generates an output signal indicative of the difference betweenthe set point of the mean flow rate and the output signal of thedifferential pressure transducer 23, and is coupled to provide thisoutput signal to the low pass filter 27. The low pass filter 27 modifiesthe output signal from the adder 26 thus generating a filtered signal.

Preferably, the low-pass filter 27 has a cut-off frequency of about 5Hz, for example. A low-frequency power amplifier 28 is coupled toreceive the filtered signal from the low-pass filter 27, and amplifiesthe filtered signal to generate an output signal. The low-frequencypower amplifier 28 is coupled to provide its output signal to the driftactuator 24 (see FIG. 7). The drift actuator 24 moves the tube 16 by anamount proportional to or based on the output-signal from thelow-frequency power amplifier 28, to control a mean area of the orifice15. Thus, the controller 25 provides the capability to compensate fordrift in the magnetostricitve actuator 18 as well as for heating orother effects that cause drift displacement of the tube 16 relative tothe mean position of tapered needle 17. The drift actuator 24 can be alinear actuator using a piezo-ceramic material that contracts orexpands, based on the output signal from the low frequency poweramplifier 28

Although the subject invention has been described with specificillustrations and embodiments, it will be clear to those of ordinaryskill in the art that various modifications may be made therein withoutdeparting from the spirit and scope of the invention as outlined in thefollowing claims. For example, although the system 1 of this inventionhas been described as controlling a combustor system to damp combustioninstabilities therein, this invention can be applied to a combustor tofurther excite combustor (or other process) oscillations by simplemodification of the processing performed in FIGS. 6A and 6B, forexample, by reversing the terms (a_(i),j)² +(b_(i),j)² !^(1/2) and(a_(i-1),j)² +(b_(i-1),j)² !^(1/2) in the inequality of step S8.Further, this invention can readily be applied to damping or enhancementof oscillations modes in turbo machinery, structures or machines. Forthese applications, the system's sensor is adapted to sense vibration inthe turbo machinery, structure or machine, and to generate a signalindicative of this vibration to the observer. As previously explained inthis document, the observer determines the modal functions and frequencyfor each mode of oscillation in the sensed signal, and the controllergenerates a control signal with a gain and phase shift, based on themodal functions and frequency for each mode. The actuator inducesvibrations in the turbo machinery, structure or machine, based on thecontrol signal generated by the controller. In addition, this inventioncan also be applied to signal identification in a noisy background,coherent jamming identification and general real time signalidentification, due to the capability of this invention to rapidlyidentify the modal functions (and hence the amplitude and phase) and thefrequency of a signal. Further, this invention can be applied to damp orenhance sound in an enclosure. Specifically, a sensor such as amicrophone can be placed in an enclosure, and its signal, generatedbased on sound waves in the enclosure, supplied to the observer of thisinvention to determine the modal functions and frequencies for each modeof oscillation. Based on the modal functions and frequencies, thecontroller of this invention generates a control signal with a gain andphase shift, that is supplied to an actuator such as a speaker. By usingan appropriate gain and phase shift, the speaker can act to enhance ordamp, as desired, sound waves in the enclosure. As will be furtherunderstood and appreciated by those possessing ordinary skill in theart, additional modifications, consistent with the concepts andteachings of the invention, may be employed. For example, the inventionmay be employed in systems utilizing a compressor and having a plenumreceiving a fluid. In such a system, the invention may be realized as amethod for real time identification of and compensation for a mode ofoscillation of the system. To accomplish the inventive aspects, themethod uses a sensor to generate a signal f(t). Thereafter, the methodoperates to determine values of modal functions a(t) and b(t) of themode of oscillation of the signal f(t), based on a pair of integralsoY_(a) f(t)dt and oY_(b) f(t)dt, respectively, Y_(a) and Y_(b) beingpredetermined wavelet functions, as previously described, and that areorthogonal and localized at an estimated frequency w of the mode ofoscillation. As discussed, the pair of integrals have limits ofintegration based on a time t and an estimated period T=2p/w of the modeof oscillation, wherein the estimated frequency w is predeterminedinitially. The method then determines an updated value of the estimatedfrequency w, based on the modal functions a(t) and b(t), and updates thewavelet functions Y_(a) and Y_(b), based on the updated value of theestimated frequency w. Thereafter, the method generates a control signalto control the oscillation in the system, based on the values of themodal functions a(t) and b(t), the wavelet functions Y_(a) and Y_(b),and the updated value of the estimated frequency w. Finally, the methodbleeds the fluid from the plenum, based on the control signal. The aboveapplications are intended to be included in the scope and spirit of thisinvention.

We claim:
 1. A method for real time identification of and compensationfor a mode of oscillation of a system, the method comprising the stepsof:a) sensing the system to generate a signal f(t); b) determiningvalues of modal functions a(t) and b(t) of the mode of oscillation ofthe signal f(t), based on a pair of integrals ∫Ψ_(a) f(t)dt and ∫Ψ_(b)f(t)dt, respectively, Ψ_(a) and Ψ_(b) being predetermined waveletfunctions that are orthogonal and localized at an estimated frequency ωof the mode of oscillation, the pair of integrals having limits ofintegration based on a time t and an estimated period T=2p/ω of the modeof oscillation, the estimated frequency ω being predetermined initially;c) determining an updated value of the estimated frequency ω, based onthe modal functions a(t) and b(t); d) updating the wavelet functionsΨ_(a) and Ψ_(b), based on the updated value of the estimated frequencyω; e) generating a control signal to control the oscillation in thesystem, based on the values of the modal functions a(t) and b(f), thewavelet functions Ψ_(a) and Ψ_(b), and the updated value of theestimated frequency ω; and, wherein the system includes a combustor, f)supplying fuel to the combustor, based on the control signal.
 2. Amethod for real time identification of and compensation for a mode ofoscillation of a system, the method comprising the steps of:a) sensingthe system to generate a signal f(t); b) determining values of modalfunctions a(t) and b(t) of the mode of oscillation of the signal f(t),based on a pair of integrals ∫Ψ_(a) f(t)dt and ∫Ψ_(b) f(t)dt,respectively, Ψ_(a) and Ψ_(b) being predetermined wavelet functions thatare orthogonal and localized at an estimated frequency ω of the mode ofoscillation, the pair of integrals having limits of integration based ona time t and an estimated period T=2p/ω of the mode of oscillation, theestimated frequency ω being predetermined initially; c) determining anupdated value of the estimated frequency ω, based on the modal functionsa(t) and b(t); d) updating the wavelet functions Ψ_(a) and Ψ_(b), basedon the updated value of the estimated frequency ω; e) generating acontrol signal to control the oscillation in the system, based on thevalues of the modal functions a(t) and b(t), the wavelet functions Ψ_(a)and Ψ_(b), and the updated value of the estimated frequency ω; and,wherein the system includes a compressor having a plenum receiving afluid, f) bleeding the fluid from the plenum, based on the controlsignal.
 3. A method for real time identification of and compensation fora mode of oscillation of a system, the method comprising the steps of:a)sensing the system to generate a signal f(t); b) determining values ofmodal functions a(t) and b(t) of the mode of oscillation of the signalf(t), based on a pair of integrals ∫Ψ_(a) f(t)dt and ∫Ψ_(b) f(t)dt,respectively, Ψ_(a) and Ψ_(b) being predetermined wavelet functions thatare orthogonal and localized at an estimated frequency ω of the mode ofoscillation, the pair of integrals having limits of integration based ona time t and an estimated period T=2p/ω of the mode of oscillation, theestimated frequency ω being predetermined initially; c) determining anupdated value of the estimated frequency ω, based on the modal functionsa(t) and b(t); d) updating the wavelet functions Φ_(a) and Ψ_(d), basedon the updated value of the estimated frequency ω; and e) replacing themodal functions a(t) and b(t) at an end of the estimated period T withaverage values A(T) and B(T) accumulated over the estimated period T ofthe mode of oscillation.
 4. A method for real time identification of themodes of oscillation of a system, the method comprising the steps of:a)initializing modal output signals y(t) for L modes of oscillation, Lbeing a predetermined integer; b) sensing the system and generating asignal indicative of the L modes of oscillation; c) combining the sensedsignal and the modal output signals y(t) of the L modes of oscillationmultiplied by respective predetermined weighting functions, to generateL modal input signals f(t); d) determining modal functions a(t) and b(t)for each of the L modes of oscillation of the sensed signal, based on Lpairs of integrals ∫Ψ_(a) f(t)dt and ∫Ψ_(b) f(t)dt, respectively, Ψ_(a)and Ψ_(b) being predetermined wavelet functions that are orthogonal andlocalized at an estimated frequency ω of each of the L modes ofoscillation, the pair of integrals having limits of integration based ona time t and an estimated period T=2p/ω of the mode of oscillation, theestimated frequencies ω being predetermined initially for each of the Lmodes of oscillation; e) determining updated values of the estimatedfrequencies ω for the L modes of oscillation, based on the modalfunctions a(t) and b(t); f) updating the wavelet functions Ψ_(a) andΨ_(b) for the L modes of oscillation, based on respective updated valuesof the estimated frequencies ω; g) generating a control signal tocontrol the oscillation in the system, based on the values of the modalfunctions a(t) and b(t), the wavelet functions Ψ_(a) and Ψ_(b) and theupdated values of the estimated frequencies ω; and, wherein the systemincludes a combustor, h) supplying fuel to the combustor, based on thecontrol signal.
 5. A method for real time identification of the modes ofoscillation of a system, the method comprising the steps of:a)initializing modal output signals y(t) for L modes of oscillation, Lbeing a predetermined integer; b) sensing the system and generating asignal indicative of the L modes of oscillation; c) combining the sensedsignal and the modal output signals Y(t) of the L modes of oscillationmultiplied by respective predetermined weighting functions, to generateL modal input signals f(t); d) determining modal functions a(t) and b(t)for each of the L modes of oscillation of the sensed signal, based on Lpairs of integrals ∫Ψ_(a) f(t)dt and ∫Ψ_(b) f(t)dt, respectively, Ψ_(a)and Ψ_(b) being predetermined wavelet functions that are orthogonal andlocalized at an estimated frequency ω of each of the L modes ofoscillation, the pair of integrals having limits of integration based ona time t and an estimated period T=2p/ω of the mode of oscillation, theestimated frequencies ω being predetermined initially for each of the Lmodes of oscillation; e) determining updated values of the estimatedfrequencies ω for the L modes of oscillation, based on the modalfunctions a(t) and b(t); f) updating the wavelet functions Ψ_(a) andΨ_(b) for the L modes of oscillation, based on respective updated valuesof the estimated frequencies ω; g) generating a control signal tocontrol the oscillation in the system, based on the values of the modalfunctions a(t) and b(t), the wavelet functions Ψ_(a) and Ψ_(b) and theupdated values of the estimated frequencies ω; and, wherein the systemincludes a compressor having a plenum receiving a fluid, h) bleeding thefluid from the plenum, based on the control signal.
 6. A method for realtime identification of the modes of oscillation of a system, the methodcomprising the steps of:a) initializing modal output signals y(t) for Lmodes of oscillation, L being a predetermined integer; b) sensing thesystem and generating a signal indicative of the L modes of oscillation;c) combining the sensed signal and the modal output signals y(t) of theL modes of oscillation multiplied by respective predetermined weightingfunctions, to generate L modal input signals f(t); d) determining modalfunctions a(t) and b(t) for each of the L modes of oscillation of thesensed signal, based on L pairs of integrals ∫Ψ_(a) f(t)dt and ∫Ψ_(b)f(t)dt, respectively, Ψ_(a) and Ψ_(b) being predetermined waveletfunctions that are orthogonal and localized at an estimated frequency ωof each of the L modes of oscillation, the pair of integrals havinglimits of integration based on a time t and an estimated period T=2p/ωof the mode of oscillation, the estimated frequencies ω beingpredetermined initially for each of the L modes of oscillation; e)determining updated values of the estimated frequencies ω for the Lmodes of oscillation, based on the modal functions a(t) and b(t); f)updating the wavelet functions Ψ_(a) and Ψ_(b) for the L modes ofoscillation, based on respective updated values of the estimatedfrequencies ω; and g) replacing the modal functions a(t) and b(t) at anend of an estimated period T for each of the L modes of oscillation withrespective average values A(T) and B(T) accumulated over a respectiveestimated period T.
 7. A system receiving a signal indicative of a modeof oscillation and a fluid flow, the system comprising:an observerincluding a processor, coupled to receive the signal, determining valuesof modal functions a(t) and b(t) of the mode of oscillation of thesignal f(t), based on a pair of integrals ∫Ψ_(a) f(t)dt and ∫Ψ_(b)f(t)dt, respectively, Ψ_(a) and Ψ_(b) being predetermined waveletfunctions that are orthogonal, localized at an estimated frequency ω ofeach of the L modes of oscillation, the pair of integrals having limitsof integration based on a time t and an estimated period T=2p/ω of themode of oscillation, the estimated frequencies ω being predeterminedinitially for each of the L modes of oscillation, the observerdetermining updated values of the estimated frequencies ω for the Lmodes of oscillation, based on the modal functions a(t) and b(t), andthe observer updating the wavelet functions Ψ_(a) and Ψ_(b) for the Lmodes of oscillation, based on respective updated values of theestimated frequencies ω; a first controller coupled to receive thevalues of the modal functions a(t) and b(t) and the frequency ωdetermined by the observer, the first controller generating a firstcontrol signal based on the modal functions a(t) and b(t), the frequencyω determined by the observer, a gain G and a phase shift f, andoutputting a first control signal; a combustor generating the mode ofoscillation based on the fluid flow; a first sensor arranged inproximity to the combustor, sensing the mode of oscillation to generatethe signal indicative of the mode of oscillation; and a first actuatorcoupled to receive the first control signal from the first controller,controlling the fluid flow into the combustor, based on the firstcontrol signal.
 8. A system as claimed in claim 7, further comprising:asecond sensor coupled to the combustor, generating a signal based on anactual flow rate of the fluid; a second controller coupled to the secondsensor and coupled to receive a signal setting a predetermined mean flowrate, the second controller generating a second control signal based onthe signal indicative of the predetermined mean flow rate and the signalindicative of the actual flow rate; and a second actuator coupled to thesecond controller, controlling a mean fluid flow based on the secondcontrol signal.